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Discontinuities, Generalized Solutions, and (Dis)agreement in Opinion Dynamics

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Control Subject to Computational and Communication Constraints

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 475))

Abstract

This chapter is devoted to the mathematical analysis of some continuous-time dynamical systems defined by ordinary differential equations with discontinuous right-hand side, which arise as models of opinion dynamics in social networks. Discontinuities originate because of specific communication constraints, namely, quantization or bounded confidence. Solutions of these systems may or may not converge to a state of agreement, where all components of the state space are equal. After presenting three models of interest, we elaborate on the properties of their solutions in terms of existence, completeness, and convergence.

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Notes

  1. 1.

    It would be more precise to write \(S_\mathbf{b}(x_0, I(x_0))\) instead of \(S_\mathbf{b}(x_0)\), as it depends on \(I(x_0)\). Note however that if \(I(x_0)\) and \(I'(x_0)\) are two distinct neighborhoods of \(x_0\), then the sets \(S_\mathbf{b}(x_0, I(x_0))\) and \(S_\mathbf{b}(x_0, I'(x_0))\) coincide on \(I(x_0)\cap I'(x_0)\). Hence, neglecting \(I(x_0)\) from the notation brings no ambiguity.

  2. 2.

    Even though the corollary is new, it could have been deduced by inspecting the proofs in [7].

  3. 3.

    Note that this is a “weak” notion of equilibrium: in case of multiple solutions, we do not require that all solutions remain at the equilibrium.

  4. 4.

    We refer the reader to [23], Chap. 1] for the relevant definitions about directed graphs.

  5. 5.

    The discretization of the opinions in social systems has been observed by social scientists [30, Chap. 10] and addressed in several models including [20, 37, 48].

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Author information

Authors and Affiliations

  1. Politecnico di Torino, c.so Duca degli Abruzzi 24, 10129, Torino, Italy

    F. Ceragioli

  2. Univ. Grenoble Alpes, CNRS, Inria, Grenoble INP, GIPSA- Lab, 38000, Grenoble, France

    P. Frasca

Authors
  1. F. Ceragioli
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  2. P. Frasca
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Correspondence to P. Frasca .

Editor information

Editors and Affiliations

  1. LAAS-CNRS, Université de Toulouse, Toulouse, France

    Sophie Tarbouriech

  2. Laboratoire des Signaux et Systèmes, CNRS, CentraleSupélec, Université Paris-Sud, Université Paris-Saclay, Gif-sur-Yvette, France

    Antoine Girard

  3. CNRS, Centrale Lille, UMR 9189—Centre de Recherche en Informatique Signal et Automatique de Lille (CRIStAL), Université Lille, Lille, France

    Laurentiu Hetel

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Ceragioli, F., Frasca, P. (2018). Discontinuities, Generalized Solutions, and (Dis)agreement in Opinion Dynamics. In: Tarbouriech, S., Girard, A., Hetel, L. (eds) Control Subject to Computational and Communication Constraints. Lecture Notes in Control and Information Sciences, vol 475. Springer, Cham. https://doi.org/10.1007/978-3-319-78449-6_14

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  • DOI: https://doi.org/10.1007/978-3-319-78449-6_14

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